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Linear functions are a fundamental concept in mathematics. They can be written in the form of \[ f(x) = ax + b \, ( \text{or} \ \frac{ax+b}{c})\] where \(a\), \(b\), and \(c\) are constants. A linear function has no bends or curves - it's a straight line when graphed.

For example, \(f(x)=\frac{3x+1}{2}\) is a linear function. The numerator \(3x+1\) is a first-degree polynomial (highest power of \(x\) is 1), and the denominator 2 is a constant. Since this function does not include any terms that might cause division by zero or taking the square root of negative numbers, it is well-defined for all real numbers.

This is what makes linear functions very accessible and easy to work with in many different areas of math and applied science.

When analyzing a function like \(f(x)=\frac{3x+1}{2}\), it's essential to recognize the components.

\(3x+1\) is the numerator and a simple linear expression. It represents how the function's value changes as \(x\) changes.

The denominator, in this case, is 2, which is a constant. Because this is not zero, the function never becomes undefined. If the denominator variable itself were a term including \(x\), we would need to check for any values that might cause division by zero.

By understanding these structures, we can predict a function's behavior. For \(f(x)=\frac{3x+1}{2}\), we have a straightforward function to understand and graph because there are no complex constraints or undefined points.

The real numbers include all the numbers you can think of that are not imaginary. This includes:

• Whole numbers (0, 1, 2, ...)
• Positive and negative integers (-1, -2, -3, ...)
• Fractions or rational numbers (1/2, 3/4, ...)
• Irrational numbers (numbers that cannot be written as fractions, like \( \pi \) and \( \sqrt{2} \))

When determining the domain of a function, we're asking: